In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

, the category of sets, denoted by Set, is the

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

whose

objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...

are sets. The arrows or

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

s between sets ''A'' and ''B'' are the

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

s from ''A'' to ''B'', and the composition of morphisms is the

composition of functions In mathematics, the composition operator \circ takes two functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is applied after applying to . (g \circ f) is pronounced "the composition of an ...

. Many other categories (such as the

category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories The ...

, with

group homomorphisms In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) whe ...

as arrows) add structure to the objects of the category of sets or restrict the arrows to functions of a particular kind (or both).

Properties of the category of sets

The axioms of a category are satisfied by Set because composition of functions is

associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...

, and because every set ''X'' has an

identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

id_''X'' : ''X'' → ''X'' which serves as identity element for function composition. The

epimorphism In category theory, an epimorphism is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \implies g_1 = g_2. Epimorphisms are categorical analo ...

s in Set are the

surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...

maps, the

monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphis ...

s are the

injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...

maps, and the

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

s are the

bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

maps. The

empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...

serves as the

initial object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element) ...

in Set with

empty function In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called ...

s as morphisms. Every

singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...

is a

terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus no

zero object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

s in Set. The category Set is complete and co-complete. The

product Product may refer to: Business * Product (business), an item that can be offered to a market to satisfy the desire or need of a customer. * Product (project management), a deliverable or set of deliverables that contribute to a business solution ...

in this category is given by the

cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

of sets. The

coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...

is given by the

disjoint union In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...

: given sets ''A''_''i'' where ''i'' ranges over some index set ''I'', we construct the coproduct as the union of ''A''_''i''× (the cartesian product with ''i'' serves to ensure that all the components stay disjoint). Set is the prototype of a

concrete category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor makes it possible to think of the objects of the category as sets with additional ...

; other categories are concrete if they are "built on" Set in some well-defined way. Every two-element set serves as a

subobject classifier In mathematics, especially in category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object ''X'' in the category correspond to the morphisms from ''X'' to Ω. In typical examples, ...

in Set. The power object of a set ''A'' is given by its

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

, and the

exponential object In mathematics, specifically in category theory, an exponential object or map object is the category theory, categorical generalization of a function space in set theory. Category (mathematics), Categories with all Product (category theory), fini ...

of the sets ''A'' and ''B'' is given by the set of all functions from ''A'' to ''B''. Set is thus a

topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notio ...

(and in particular

cartesian closed In category theory, a Category (mathematics), category is Cartesian closed if, roughly speaking, any morphism defined on a product (category theory), product of two Object (category theory), objects can be naturally identified with a morphism defin ...

and exact in the sense of Barr). Set is not

abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a group ...

additive Additive may refer to: Mathematics * Additive function, a function in number theory * Additive map, a function that preserves the addition operation * Additive set-function see Sigma additivity * Additive category, a preadditive category with fin ...

nor preadditive. Every non-empty set is an

injective object In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. ...

in Set. Every set is a

projective object In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object ...

in Set (assuming the

axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...

). The finitely presentable objects in Set are the finite sets. Since every set is a

direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...

of its finite subsets, the category Set is a locally finitely presentable category. If ''C'' is an arbitrary category, the contravariant functors from ''C'' to Set are often an important object of study. If ''A'' is an object of ''C'', then the functor from ''C'' to Set that sends ''X'' to Hom_''C''(''X'',''A'') (the set of morphisms in ''C'' from ''X'' to ''A'') is an example of such a functor. If ''C'' is a

small category In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...

(i.e. the collection of its objects forms a set), then the contravariant functors from ''C'' to Set, together with natural transformations as morphisms, form a new category, a

functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object i ...

known as the category of

presheaves In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the da ...

on ''C''.

Foundations for the category of sets

Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...

the collection of all sets is not a set; this follows from the

axiom of foundation In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the axio ...

. One refers to collections that are not sets as

proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...

es. One cannot handle proper classes as one handles sets; in particular, one cannot write that those proper classes belong to a collection (either a set or a proper class). This is a problem because it means that the category of sets cannot be formalized straightforwardly in this setting. Categories like Set whose collection of objects forms a proper class are known as large categories, to distinguish them from the small categories whose objects form a set. One way to resolve the problem is to work in a system that gives formal status to proper classes, such as NBG set theory. In this setting, categories formed from sets are said to be ''small'' and those (like Set) that are formed from proper classes are said to be ''large''. Another solution is to assume the existence of

Grothendieck universe In mathematics, a Grothendieck universe is a set ''U'' with the following properties: # If ''x'' is an element of ''U'' and if ''y'' is an element of ''x'', then ''y'' is also an element of ''U''. (''U'' is a transitive set.) # If ''x'' and ''y'' ...

s. Roughly speaking, a Grothendieck universe is a set which is itself a model of ZF(C) (for instance if a set belongs to a universe, its elements and its powerset will belong to the universe). The existence of Grothendieck universes (other than the empty set and the set

V_\omega

of all

hereditarily finite set In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to t ...

s) is not implied by the usual ZF axioms; it is an additional, independent axiom, roughly equivalent to the existence of

strongly inaccessible cardinal In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible cardinal if it is uncountable, regular, and a weak limit cardinal. Since abou ...

s. Assuming this extra axiom, one can limit the objects of Set to the elements of a particular universe. (There is no "set of all sets" within the model, but one can still reason about the class ''U'' of all inner sets, i.e., elements of ''U''.) In one variation of this scheme, the class of sets is the union of the entire tower of Grothendieck universes. (This is necessarily a

, but each Grothendieck universe is a set because it is an element of some larger Grothendieck universe.) However, one does not work directly with the "category of all sets". Instead, theorems are expressed in terms of the category Set_''U'' whose objects are the elements of a sufficiently large Grothendieck universe ''U'', and are then shown not to depend on the particular choice of ''U''. As a foundation for

, this approach is well matched to a system like

Tarski–Grothendieck set theory Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiom ...

in which one cannot reason directly about proper classes; its principal disadvantage is that a theorem can be true of all Set_''U'' but not of Set. Various other solutions, and variations on the above, have been proposed. The same issues arise with other concrete categories, such as the

or the

category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...

Notes

References

* * *Lawvere, F.W
An elementary theory of the category of sets (long version) with commentary
* * *

External links

A231344 Number of morphisms in full subcategories of Set spanned by
at

OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...

. {{Foundations-footer Foundations of mathematics Sets Basic concepts in set theory

Properties of the category of sets

Foundations for the category of sets

See also

Notes

References

External links